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Theorem tsrlin 14653
Description: A toset is a linear order. (Contributed by Mario Carneiro, 9-Sep-2015.)
Hypothesis
Ref Expression
istsr.1  |-  X  =  dom  R
Assertion
Ref Expression
tsrlin  |-  ( ( R  e.  TosetRel  /\  A  e.  X  /\  B  e.  X )  ->  ( A R B  \/  B R A ) )

Proof of Theorem tsrlin
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 istsr.1 . . . . 5  |-  X  =  dom  R
21istsr2 14652 . . . 4  |-  ( R  e.  TosetRel 
<->  ( R  e.  PosetRel  /\  A. x  e.  X  A. y  e.  X  (
x R y  \/  y R x ) ) )
32simprbi 452 . . 3  |-  ( R  e.  TosetRel  ->  A. x  e.  X  A. y  e.  X  ( x R y  \/  y R x ) )
4 breq1 4217 . . . . 5  |-  ( x  =  A  ->  (
x R y  <->  A R
y ) )
5 breq2 4218 . . . . 5  |-  ( x  =  A  ->  (
y R x  <->  y R A ) )
64, 5orbi12d 692 . . . 4  |-  ( x  =  A  ->  (
( x R y  \/  y R x )  <->  ( A R y  \/  y R A ) ) )
7 breq2 4218 . . . . 5  |-  ( y  =  B  ->  ( A R y  <->  A R B ) )
8 breq1 4217 . . . . 5  |-  ( y  =  B  ->  (
y R A  <->  B R A ) )
97, 8orbi12d 692 . . . 4  |-  ( y  =  B  ->  (
( A R y  \/  y R A )  <->  ( A R B  \/  B R A ) ) )
106, 9rspc2v 3060 . . 3  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A. x  e.  X  A. y  e.  X  ( x R y  \/  y R x )  ->  ( A R B  \/  B R A ) ) )
113, 10syl5com 29 . 2  |-  ( R  e.  TosetRel  ->  ( ( A  e.  X  /\  B  e.  X )  ->  ( A R B  \/  B R A ) ) )
12113impib 1152 1  |-  ( ( R  e.  TosetRel  /\  A  e.  X  /\  B  e.  X )  ->  ( A R B  \/  B R A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 359    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   class class class wbr 4214   dom cdm 4880   PosetRelcps 14626    TosetRel ctsr 14627
This theorem is referenced by:  tsrlemax  14654  ordtrest2lem  17269  ordthauslem  17449  ordthaus  17450
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-xp 4886  df-rel 4887  df-cnv 4888  df-dm 4890  df-tsr 14632
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